The Unconditional Uniqueness Of Solution For One Type Nonlinear Schr(?)dinger Equation

This paper mainly discusses the unconditional uniqueness of solution for one type nonlinear Schr(?)dinger equation#12 where F(u)=?1Vu+?2|u|?+?3(?*|u|2),?i?C,i=1,2,3,with only one ?i being zero,T>0,s>0,0<?<min{4/n-2s,n+2s/n-2s},??Lw?/n(Rn),0<?<min{2s+2,n},V?Lp(Rn),1?p??.?and*represent the Laplacian operator and convolution on Rn respectively.We show the proof of the unconditional uniqueness for above equation in Sobolev space.Especially,our proof applies the negative order Sobolev space,nonhomogeneous Strichartz estimates,and the improved regularity property for the difference between two solutions of the equation.The thesis is consisted of four chapters.In the first chapter,it mainly elaborates the historical background of the nonlinear Schrodinger equation and the conclusion of the wellposedness as well as uniqueness.On this basis,we introduce the main research contents of this thesis.The second chapter mainly describes the main nonation and preparation of this thesis,which provide a theoretical foundation for the research of the main results.The third chapter gives the proofs of all theorems:The first section of Chapter 3 mainly uses Lemma 2.2.1 classical Strichartz estimates to prove Theorems 1.1 and 1.2.Ten proof of Theorem 1.2 is divided into many cases.The second section of Chapter 3 mainly uses Lemma 2.2.2 inhomogeneous Strichartz estimates to prove Theorems 1.3 and 1.4.The third section of Chapter 3 mainly uses Lemma 2.2.1 and classical Strichartz estimates to prove Theorem 1.5.According to the selection of q,r,? and ?,the proof is divided into two cases:n=1,2 and n?3.The fourth chapter makes a prospect for the next research work.Because the research work in this article is still in its infancy,it is still relatively superficial.And due to time and capacity constraints,the unconditional uniqueness of nonlinear Schrodinger solution for powertype,and some linear-type still have some unresolved problems,and these problems need to be studied in depth.